Bimodules and branes in deformation quantization

Abstract

We prove a version of Kontsevich's formality theorem for two subspaces (branes) of a vector space X. The result implies, in particular, that the Kontsevich deformation quantizations of S(X * ) and ∧(X) associated with a quadratic Poisson structure are Koszul dual. This answers an open question in Shoikhet's recent paper on Koszul duality in deformation quantization.

Biquantization of symmetric pairs and the quantum shift

“…The biquantization of symmetric pairs was studied in [7] in terms of Kontsevich-like graphs. This paper, also in view of recent results in [4], amends a minor mistake that did not spoil the main results of the paper. The mistake consisted in ignoring a regular term in the boundary contribution of some propagators.…”

“…The whole construction of [7, Section 1.6] relies on the 4-colored propagators introduced in [5] for the Poisson sigma model with two branes. It was recently observed by G. Felder and the second author in the preparation of [4], that, unlike in Kontsevich [13], the boundary contributions of the 4-colored propagators on the first quadrant for the collapse of the two endpoints may have a regular term in addition to the usual singular one. The regular term turns out simply to be the differential of the angle of the position where the two points collapsed, measured with respect to the origin, up to a sign, which depends on the boundary conditions themselves (roughly speaking, if we consider the same boundary conditions on the two half-lines bounding the first quadrant, then the sign is positive, while, for different boundary conditions on the two half-lines, we have a negative sign).…”

“…Its net effect is that more boundary contributions have to be taken into account and extra terms are needed for cancellation. It turns out [4] that it is enough to allow for the presence of short loops and to assign each of them the regular term. This also has the pleasant effect of restoring the quantum shift.…”

“…

Dans [7] la biquantification des paires symétriques aétéétudiéà l'aide du formalisme graphique de M. Kontsevich. Dans ce papier on corrige, compte tenu des résultats montrés récemment dans [4], une erreur mineure dans [7], qui en tout cas n'invalide pas les résultats plus importants dans [7] : cette erreur consiste au fait que les auteurs avaient oublié une contribution provenante d'une certaine composante du bord dans le propagateurà quatre couleurs. La correction qu'on apporte ici a l'avantage de remettre finalement en jeu la "translation quantique" des charactères qui apparaît dans la méthode des orbites, et quiétait mystèrieusement absente dans [7].

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The explicit equivalence between the standard and the logarithmic star product for Lie algebras, I

“…We denote by C on K̵ h can be obtained from the ones for the corresponding A ∞ -structure on K̵ h constructed in [4,Section 7] by replacing ω +,− by its logarithmic counterpart (5): also for later computations, we frequently and implicitly refer to [4,Section 7].…”